Developed compact multiplicative product operations and closures for collections in the leanprover-community/mathlib4 repository, focusing on formal verification and advanced mathematics. The work introduced a HasCompactMulSupport submonoid and extended product operations for lists, multisets, and finite sets, including to_additive variants, all under minimal type-class requirements. This feature reduced boilerplate and improved API coverage for algebraic and topological structures, aligning with Lean community standards. Implemented entirely in Lean, the contribution accelerates rigorous proofs involving finite-collection products and expands reusable abstractions for compact support, enhancing maintainability and extensibility of Mathlib’s algebraic and topological toolkit without introducing new bugs.